Phase estimation with randomized hamiltonians

ABSTRACT

Existing methods for dynamical simulation of physical systems use either a deterministic or random selection of terms in the Hamiltonian. In this application, example approaches are disclosed where the Hamiltonian terms are randomized and the precision of the randomly drawn approximation is adapted as the required precision in phase estimation increases. This reduces both the number of quantum gates needed and in some cases reduces the number of quantum bits used in the simulation.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.62/819,301 entitled “PHASE ESTIMATION WITH RANDOMIZED HAMILTONIANS” andfiled on Mar. 15, 2019, which is hereby incorporated herein by referencein its entirety.

FIELD

This application relates generally to quantum computing. In more detail,example approaches are disclosed where Hamiltonian terms are randomizedand the precision of the randomly drawn approximation is adapted as therequired precision in phase estimation increases.

SUMMARY

Existing methods for dynamical simulation of physical systems use eithera deterministic or random selection of terms in the Hamiltonian. In thisdisclosure, example approaches are disclosed where the Hamiltonian termsare randomized and the precision of the randomly drawn approximation isadapted as the required precision in phase estimation increases. Thisreduces both the number of quantum gates needed and in some casesreduces the number of quantum bits used in the simulation.

Embodiments comprise randomizing phase estimation by replacing theHamiltonian with a randomly generated one each time it is simulated.Further embodiments involve the use of randomization within an iterativephase estimation algorithm to select Hamiltonian terms for inclusion inthe approximation as well as their ordering. Certain embodiments involvethe use of importance functionals based on the significance of the termin the groundstate to determine whether it gets included in the randomlysampled Hamiltonian. Further embodiments involve the use of importancesampling based on the variational approximations to the groundstates,such as but not limited to, CISI) states. Certain embodiments involvethe use of adaptive Bayesian methods in concert with this process toquantify the precision of the Hamiltonian needed given the currentuncertainty in the eigenvalue that the algorithm is estimating.

In this application, example methods for performing a quantum simulationusing adaptive Hamiltonian randomization. The particular embodimentsdescribed should not be construed as limiting, as the disclosed methodacts can be performed alone, in different orders, or at least partiallysimultaneously with one another. Further, any of the disclosed methodsor method acts can be performed with any other methods or method actsdisclosed herein. In particular embodiments, a Hamiltonian to becomputed by the quantum computer device is inputted; a number ofHamiltonian terms in the Hamiltonian is reduced using randomizationwithin a phase estimation algorithm; and a quantum circuit descriptionfor the Hamiltonian is output with the reduced number of Hamiltonianterms.

In certain embodiments, the reducing comprises selecting one or morerandom Hamiltonian terms based on an importance function; reweightingthe selected random Hamiltonian terms based on an importance of each ofthe selected Hamiltonian random terms; and generating the quantumcircuit description using the reweighted random terms. Some embodimentsfurther comprise implementing, in the quantum computing device, aquantum circuit as described by the quantum circuit description; andmeasuring a quantum state of the quantum circuit. Still furtherembodiments comprise re-performing the method based on results from themeasuring (e.g., using an iterative process). In some embodiments, theiterative process comprises computing a desired precision value for theHamiltonian; computing a standard deviation for the Hamiltonian based onresults from the implementing and measuring; and comparing the desiredprecision value to the standard deviation. Some embodiments furthercomprise changing an order of the Hamiltonian terms based on thereducing. Certain embodiments further comprise applying importancefunctions to terms of the Hamiltonian in a ground state; and selectingone or more random Hamiltonian terms based at least in part on theimportance functions. Some embodiments comprise using importancesampling based on a variational approximation to a groundstate. Certainembodiments further comprise using adaptive Bayesian methods to quantifya precision needed for the Hamiltonian given an estimate of the currentuncertainty in an eigenvalue.

Other embodiments comprise one or more computer-readable media storingcomputer-executable instructions, which when executed by a computercause the computer to perform a method comprising inputting aHamiltonian to be computed by the quantum computer device; reducing anumber of Hamiltonian terms in the Hamiltonian using randomizationwithin a phase estimation algorithm; and outputting a quantum circuitdescription for the Hamiltonian with the reduced number of Hamiltonianterms.

The foregoing and other objects, features, and advantages of thedisclosed technology will become more apparent from the followingdetailed description, which proceeds with reference to the accompanyingfigures.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a quantum circuit for performing iterative phaseestimation.

FIGS. 2-9 comprise graphs that show the average ground energy shift(compared to unsampled Hamiltonian), variance in ground energies oversampled Hamiltonians, average qubit requirement, and average number ofterms in sampled Hamiltonians for Li₂, as a function of number ofsamples taken to generate the Hamiltonian and the value of the parameterp.

FIG. 10 is a flow chart showing an example method for implementing animportance sampling simulation method according to an embodiment of thedisclosed technology.

FIG. 11 is a flow chart showing an example method for performing aquantum simulation using adaptive Hamiltonian randomization.

FIG. 12 illustrates a generalized example of a suitable classicalcomputing environment in which aspects of the described embodiments canbe implemented.

FIG. 13 shows an example of a possible network topology (e.g., aclient-server network) for implementing a system according to thedisclosed technology.

FIG. 14 shows another example of a possible network topology (e.g., adistributed computing environment) for implementing a system accordingto the disclosed technology.

FIG. 15 shows an exemplary system for implementing the disclosedtechnology.

FIG. 16 is a flow chart showing an example method for performing aquantum simulation using adaptive Hamiltonian randomization.

DETAILED DESCRIPTION I. Introduction

Not all Hamiltonian terms are created equally in quantum simulation.Hamiltonians that naturally arise from chemistry, materials and otherapplications often are composed of terms that are negligibly small.These terms are often culled from the Hamiltonian well before it hitsthe simulator. Other terms that are formally present in the Hamiltonianare removed, not because of their norm, but rather because they are notexpected to impact the quantity of interest. For example, in quantumchemistry one usually selects an active space of orbitals and ignoresany orbitals outside the active space. This causes many large terms tobe omitted from the Hamiltonian.

This process, often called decimation, often involves systematicallyremoving terms from the Hamiltonian and simulating the dynamics. Theidea behind such a scheme is to remove terms in the Hamiltonian untilthe maximum shift allowed in the eigenvalues is comparable to the levelof precision needed. For the case of chemistry, chemical accuracy sets anatural accuracy threshold for such simulations, but in general thisprecision requirement need not be viewed as a constant.

One of the example innovations of this disclosure is that, in iterativephase estimation, the number of terms taken in the Hamiltonian shouldideally not be held constant. The reason why is that the high-order bitsare mostly irrelevant when one is trying to learn, for example, a givenbit of a binary expansion of the eigenphase. A much lower accuracysimulation can be tolerated than it can when learning a high-order bit.It then makes sense to adapt the number of terms in the Hamiltonian asiterative phase estimation proceeds through the bits of the phaseestimation. Example embodiments of the disclosed technology provide asystematic method for removing terms and provides formal proofs thatsuch processes need not dramatically affect the results of phaseestimation nor its success probability.

One of the concepts behind the example advanced decimation procedure isthat a form of importance sampling is used to estimate, a priori, whichterms in the Hamiltonian are significant. These randomized Hamiltoniansare then used within a simulation circuit to prepare approximate groundstates. It is then shown that, using analysis that is reminiscent ofthat behind the Zeno effect or the quantum adiabatic theorem, that theerrors in the eigenstate prepared at each round of phase estimation neednot have a substantial impact on the posterior mean of the eigenphaseestimated for the true Hamiltonian. This shows, under appropriateassumptions on the eigenvalue gaps, that this process can be used toreduce the time complexity of simulation and even under somecircumstances reduce the space complexity by identifying qubits that arenot needed for the level of precision asked of the simulation.

The disclosure proceeds by first reviewing iterative phase estimationand Bayesian inference, which are used to quantify the maximal error inthe inference of the phase. The disclosure then proceeds to examine theeffect of using a stochastic Hamiltonian on the eigenphases yielded byphase estimation in the simple case where a fixed, but random,Hamiltonian is used at each step of iterative phase estimation. The morecomplicated case is then examined where each repetition of e^(−iHt) inthe iterative phase estimation circuit is implemented with a differentrandom Hamiltonian. The theoretical analysis concludes by showing thatthe success probability is not degraded substantially if the eigenvaluegaps of the original Hamiltonian are sufficiently large. Further,numerical examples of this sampling procedure are shown and from that itcan be concluded that the example sampling process for the Hamiltoniancan have a substantial impact on the number of terms in the Hamiltonianand even in some cases the number of qubits used in the simulation.

II. Iterative Phase Estimation

The idea behind iterative phase estimation is based on the aim to builda quantum circuit that acts as an interferometer wherein the unitary onewishes to probe is applied in one of the two branches of theinterferometer but not applied in the other. When the quantum state isallowed to interfere with itself at the end of the protocol theinterference pattern reveals the eigenphase. This process allows theeigenvalues of U to be estimated within the standard quantum limit (i.e.the number of applications of U needed to estimate the phase withinerror ϵ is in Θ(1/ϵ²). If the quantum state is allowed to passrepeatedly through the interferometer circuit (or entangled inputs areused) then this scaling can be reduced to Θ(1/ϵ) which is known as theHeisenberg limit. Such a circuit is shown in schematic block diagram 100of FIG. 1. In particular, FIG. 1 shows a quantum circuit for performingiterative phase estimation. M is the number of repetitions of theunitary U (not necessarily an integer), and θ is a phase offset betweenthe ancilla |0

and |1

states.

The phase estimation circuit is easy to analyze in the case where U|ψ

=e^(iϕ)|ψ

. If U is repeated M times and θ is a phase offset then the likelihoodof a given measurement outcome oϵ{0, 1} for the circuit in FIG. 1 withthese parameters is

$\begin{matrix}{{\Pr \left( {{{o\varphi};M},\theta} \right)} = {\frac{1 + {\left( {- 1} \right){{{^\circ}cos}\left( {M\left( {\theta - \varphi} \right)} \right)}}}{2}.}} & (1)\end{matrix}$

There are many free parameters that can be used when designing iterativephase estimation experiments. In particular, the rules for generating Mand θ for each experiment vary radically along with the methods used toprocess the data that comes back from these experiments. Approaches suchas Kitaev's phase estimation algorithm, robust phase estimation,information theory phase estimation, or any number of approximateBayesian methods, provide good heuristics for picking these parameters.In this disclosure, it is assumed that one does not wish to specify toany of these methods for choosing experiments, nor does one wish tofocus on the specific data processing methods used. Nonetheless,Bayesian methods are relied on to discuss the impact that randomizingthe Hamiltonian can have on an estimate of the eigenphase.

Bayes' theorem can be interpreted as giving the correct way to updatebeliefs about some fact given a set of experimental evidence and priorbeliefs. The initial beliefs of the experimentalist are encoded by aprior distribution Pr(ϕ). In many cases, it is appropriate to set Pr(ϕ)to be a uniform distribution on [0, 2π) to represent a state of maximalignorance about the eigenphase. However, in quantum simulation broaderpriors can be chosen if each step in phase estimation usesU_(j)=e^(−iHt) ^(j) and obeys U_(j)|ψ

=e^(−iE) ⁰ ^(t) ^(j) |ψ

for different t_(j), since such experiments can learn E₀ as opposed toexperiments with a fixed t which yield ϕ=E₀t mod 2π.

Bayes' theorem then gives the posterior distribution Pr(ϕ|o; ϕ, M) to be

$\begin{matrix}{{\Pr \left( {{{\varphi o};\varphi},M} \right)} = {\frac{{\Pr \left( {{{o\varphi};M},\theta} \right)}{\Pr (\varphi)}}{\int{{\Pr \left( {{{o\varphi};M},\theta} \right)}{\Pr (\varphi)}d\; \varphi}}.}} & (2)\end{matrix}$

Given a complete data set rather than a single datum, one has that

$\begin{matrix}{{\Pr \left( {{{\overset{\rightarrow}{\varphi}\overset{\rightarrow}{o}};\varphi},\overset{\rightarrow}{M}} \right)} = {\frac{\overset{\;}{\prod_{j}}{{\Pr \left( {{{o_{j}\varphi};M_{j}},\theta_{j}} \right)}{\Pr (\varphi)}}}{\int{\prod_{j}{{\Pr \left( {{{o_{j}\varphi};M_{j}},\theta_{j}} \right)}{\Pr (\varphi)}d\; \varphi}}}.}} & (3)\end{matrix}$

This probability distribution encodes the experimentalist's entire stateof knowledge about ϕ given that the data is processed optimally.

It is not customary to return the posterior distribution (or anapproximation thereof) as output from a phase estimation protocol.Instead, a point estimate for ϕ is given. The most frequently usedestimate is the maximum a posteriori (MAP) estimate which is simply thed that has the maximum probability. While this quantity has a niceoperational interpretation, it suffers from a number of deficiencies forpurposes of this disclosure. The main drawback here is that the MAPestimate is not robust, in the sense that if two different values of ϕhave comparable likelihoods then small errors in the likelihood can leadto radical shifts in the MAP estimate. The posterior mean is a betterestimate for this purpose, which formally is ∫ Pr(ϕ|{right arrow over(o)}; {right arrow over (M)}, {right arrow over (θ)})ϕdϕ. The posteriormean can be seen as the estimate that reduces the mean square error inany unbiased estimate of ϕ and thus it is well motivated. It also hasthe property that it is robust to small perturbations in the likelihood,which is a feature that is used below to estimate the impact on theresults of a phase estimation experiment.

III. Errors in Likelihood Function A. Linear Combinations of Unitaries

Linear combination of unitary methods for quantum simulation haverapidly become a favored method for simulating Hamiltonian dynamics inquantum systems. Unlike Trotter decompositions, many of these methods donot necessarily yield a unitary approximation to the simulated quantumdynamics. This means that it is impossible to use Stone's theoremdirectly to argue that the linear combination of unitaries methodimplements e^(−i{tilde over (H)}t) in place of e^(−iHt). In turn, sincethe standard analysis of the error propagation from Trotterdecompositions to the estimated phase in iterative phase estimationfails because one cannot reason about the eigenvalues of {tilde over(H)} directly.

Here, to address this in part, a discussion is provided of the impactthat such errors can have on the likelihood function for iterative phaseestimation.

Lemma 1. Let V be a possibly non-unitary operation that can benon-deterministically performed by a quantum computer in a controlledfashion such that there exists a unitary U with ∥U−V∥≤δ<1. If onedefines the likelihood function post-selected on V succeeding to be{tilde over (P)}(o|Et; M, θ) then

${{{P\left( {{{o{Et}};M},\theta} \right)} - {\overset{\sim}{P}\left( {{{o{Et}};M},\theta} \right)}}} \leq {\frac{\delta}{1 - \delta}.}$

Proof. Let one assume that o=0. Then, one has that for input state |ψ

the error in the likelihood function output by iterative phaseestimation is

$\begin{matrix}{{{{P\left( {{{0{Et}};M},\theta} \right)} - {\overset{\sim}{P}\left( {{{0{Et}};M},\theta} \right)}}} = {{{{{Tr}\left( {\left\lbrack \frac{{{Ue}^{i\; \theta}{\psi\rangle}\; {\langle\psi }} + {e^{{- i}\; \theta}{\psi\rangle}\; {\langle\psi }U^{\dagger}}}{4} \right\rbrack - \left\lbrack \frac{{{Ve}^{i\; \theta}{\psi\rangle}\; {\langle\psi }} + {e^{{- i}\; \theta}{\psi\rangle}\; {\langle\psi }V^{\dagger}}}{4{{V{\psi\rangle}}}} \right\rbrack} \right)}} \leq {\frac{1}{2}{{{Tr}\left( {{U{\psi\rangle}\; {\langle\psi }} - \frac{V{\psi\rangle}\; {\langle\psi }}{{V{\psi\rangle}}}} \right)}}} \leq {{\frac{1}{2}{{{Tr}\left( {{U{\psi\rangle}\; {\langle\psi }} - {V{\psi\rangle}\; {\langle\psi }}} \right)}}} + {\frac{1}{2}{{{Tr}\left( {{V{\psi\rangle}\; {\langle\psi }} - \frac{V{\psi\rangle}\; {\langle\psi }}{{V{\psi\rangle}}}} \right)}}}} \leq {\frac{\delta}{2} + {\frac{V}{2}\left( {\frac{1}{1 - \delta} - 1} \right)}} \leq {\frac{\delta}{2} + {\frac{1 + \delta}{2}\left( \frac{\delta}{1 - \delta} \right)}}} = {\frac{\delta}{1 - \delta}.}}} & (4)\end{matrix}$

Since P(0|Et; M, θ)+P(1|Et; M, θ)=1 it follows that the same bound mustapply for o=1 as well. Thus the result holds for any o as claimed.

This result, while straight forward, is significant because it allowsthe maximum errors in the mean of the posterior distribution to bepropagated through the iterative phase estimation protocol. The abilityto propagate these errors will ultimately allow us to show thatiterative phase estimation can be used to estimate eigenvalues fromlinear combinations of unitary methods.

B. Subsampling Hamiltonians

The case where terms are sampled uniformly from the Hamiltonian is nowconsidered. Let the Hamiltonian be a sum of L simulable HamiltoniansH_(l), H

, Throughout, an eigenstate |ψ

of H and its corresponding eigenenergy E is considered. From theoriginal, one can construct a new Hamiltonian

$\begin{matrix}{H_{est} = {\frac{L}{m}{\sum\limits_{i = 1}^{m}H_{_{i}}}}} & (5)\end{matrix}$

by uniformly sampling terms H_(l) _(i) from the original Hamiltonian.

When one randomly sub-samples the Hamiltonian, errors are naturallyintroduced. The main question is less about how large such errors are,but rather how they impact the iterative phase estimation protocol. Thefollowing lemma states that the impact on the likelihood functions canbe made arbitrarily small.

Lemma 2. Let

_(i) be an indexed family of sequences mapping {1, . . . . , m}→{1, . .. , L} formed by uniformly sampling elements from {1, . . . , L}independently with replacement and let {H_(l):

=1, . . . , L} be a corresponding family of Hamiltonians with H=

. For |ψ

an eigenstate of H such that H|ψ

=E|ψ

and

$H_{samp} = {\frac{L}{m}{\sum\limits_{k = 1}^{m}H_{_{i}{(k)}}}}$

with corresponding eigenstate H_(samp)|ψ_(i)

=E_(i)|ψ_(i)

one then has that the error in the likelihood function for phaseestimation vanishes with high probability over H_(samp) in the limit oflarge m:

${{{P\left( {{{o{Et}};M},\theta} \right)} - {P\left( {{{o{E_{i}t}};M},\theta} \right)}}} \in {O\mspace{11mu} \left( {\frac{MtL}{\sqrt{m}}\sqrt{_{}\left( {{\langle\psi }H_{}{\psi\rangle}} \right)}} \right)}$

Proof. Because the terms H_(l) _(i) _((k)) are uniformly sampled, eachset of terms {

_(i)} is equally likely, and by linearity of expectation

[H_(samp)]=H, from which one knows that

_({i}) [

ψ|H−H_(est)|ψ

]=0.

The second moment is easy to compute from the independence property ofthe distribution.

$\begin{matrix}{{_{\{ i\}}\left( {{\langle\psi }H_{samp}{\psi\rangle}} \right)} = {{\frac{L^{2}}{m^{2}}{_{\{ i\}}\left( {{\langle\psi }{\sum\limits_{k = 1}^{m}{H_{_{i}{(k)}}{\psi\rangle}}}} \right)}} = {\frac{L^{2}}{m^{2}}{\sum\limits_{k = 1}^{m}{{_{\{ i\}}\left( {{\langle\psi }H_{_{i}{(k)}}{\psi\rangle}} \right)}.}}}}} & (6)\end{matrix}$

Since the different

_(i) are chosen uniformly at random the result then follows from theobservation that

_({i})(

ψ|

|ψ

)=

(

ψ|

|ψ

).

From first order perturbation theory, one has that the leading ordershift in any eigenvalue is O(

ψ|(H−H_(samp))|ψ

) to within error O(L/√{square root over (m)}). Thus the variance inthis shift is from Eq. (6)

$\begin{matrix}{{_{\{ i\}}\left( {{\langle\psi }\left( {H - H_{samp}} \right){\psi\rangle}} \right)} = {\frac{L^{2}}{m}{{_{}\left( {{\langle\psi }H_{}{\psi\rangle}} \right)}.}}} & (7)\end{matrix}$

This further implies that the perturbed eigenstate |ψ_(i)

has eigenvalue

$\begin{matrix}{{H_{samp}{\psi_{i}\rangle}} = {{E{\psi_{i}\rangle}} + {O\left( {\frac{L}{\sqrt{m}}\sqrt{_{}\left( {{\langle\psi }H_{}{\psi\rangle}} \right)}} \right)}}} & (8)\end{matrix}$

with high probability over i from Markov's inequality. It then followsfrom Taylor's theorem and Eq. (1) that

$\begin{matrix}{{{{{P\left( {{{o{Et}};M},\theta} \right)} - {P\left( {{{o{E_{i}t}};M},\theta} \right)}}}\; \in {O\left( {\frac{MtL}{\sqrt{m}}\sqrt{_{}\left( {{\langle\psi }H_{}{\psi\rangle}} \right)}} \right)}},} & (9)\end{matrix}$

with high probability over i.

This result shows that if one samples the coefficients of theHamiltonian that are to be included in the sub-sampled Hamiltonianuniformly then one can make the error in the estimate of the Hamiltonianarbitrarily small. In this context, taking m→∞ does not cause the costof simulation to diverge (as it would for many sampling problems). Thisis because once every possible term is included in the Hamiltonian,there is no point in sub-sampling and one may as well take H_(samp) tobe II to eliminate the variance in the likelihood function that wouldarise from sub-sampling the Hamiltonian. In general, one needs to takemϵΩ(

(

ψ|

|ψ

)/(MtL)²) in order to guarantee that the error in the likelihoodfunction is at most a constant. Thus, this shows that as any iterativephase estimation algorithm proceeds, that (barring the problem ofaccidentally exciting a state due to perturbation) one will be able tofind a good estimate of the eigenphase by taking m to scale inversequadratically with M.

IV. Bayesian Phase Estimation Using Random Hamiltonians

Theorem 3. Let E be an event and let P(E|θ) and P′(E|θ) for θϵ[−π, π) betwo likelihood functions such that max_(θ)(|P(E|θ)−P′(E|θ)|)≤Δ andfurther assume that for prior P(θ) one has that min(P(E), P′(E))≥2Δ. Onethen has that

${{\int{{\theta \left( {{P\left( {\theta E} \right)} - {P^{\prime}\left( {\theta E} \right)}} \right)}d\; \theta}}} \leq {{\frac{5\; \pi \; \Delta}{P(E)}.{and}}\mspace{14mu} {if}}$${P\left( {E\theta} \right)} = {{{\prod\limits_{j = 1}^{N}{{P\left( {E_{j}\theta} \right)}\mspace{14mu} {with}\mspace{14mu} 1}} - {{{P^{\prime}\left( {E_{j}\theta} \right)}/{P\left( {E_{j}\theta} \right)}}}} \leq {\gamma \mspace{14mu} {then}}}$∫θ(P(θE) − P^(′)(θE))d θ ≤ 5 π((1 + γ)^(N) − 1).

Proof. From the triangle inequality, one has that

|P(E)−P′(E)|=|∫P(θ)(P(E|θ)−P′(E|θ))dθ|≤Δ.  (10)

Thus it follows from the assumption that P′(E)≥2Δ that

                                          (11) $\begin{matrix}{{{{P\left( {\theta E} \right)} - {P^{\prime}\left( {\theta E} \right)}}} = {{P(\theta)}{{\frac{P\left( {E\theta} \right)}{P(E)} - \frac{P^{\prime}\left( {E\theta} \right)}{P^{\prime}(E)}}}}} \\{\leq {{P(\theta)}\left( {{{\frac{P\left( {E\theta} \right)}{P(E)} - \frac{P^{\prime}\left( {E\theta} \right)}{P(E)}}} + {{\frac{P^{\prime}\left( {E\theta} \right)}{P(E)} - \frac{P^{\prime}\left( {E\theta} \right)}{P^{\prime}(E)}}}} \right)}} \\{\leq {{P(\theta)}\left( {\frac{\Delta}{P(E)} + {{\frac{P^{\prime}\left( {E\theta} \right)}{{P^{\prime}(E)} - \Delta} - \frac{P^{\prime}\left( {E\theta} \right)}{P^{\prime}(E)}}}} \right)}} \\{\leq {\Delta \left( {\frac{P(\theta)}{P(E)} + \frac{2{P^{\prime}\left( {E\theta} \right)}}{P^{\prime \; 2}(E)}} \right)}}\end{matrix}$

Thus one has that

$\begin{matrix}\begin{matrix}{{{{\int{{\theta \left( {{P\left( {\theta E} \right)} - {P^{\prime}\left( {\theta E} \right)}} \right)}d\; \theta}}} \leq {\Delta \left( {\frac{{\langle\theta\rangle}_{prior}}{P(E)} + \frac{2{\langle\theta\rangle}_{posterior}^{\prime}}{P^{\prime}(E)}} \right)}},} \\{\leq {\pi \; {\Delta \left( {\frac{1}{P(E)} + \frac{2}{P^{\prime}(E)}} \right)}}} \\{\leq {\pi \; {\Delta \left( {\frac{1}{P(E)} + \frac{2}{{P(E)} - \Delta}} \right)}}} \\{\leq {\frac{5\; \pi \; \Delta}{P(E)}.}}\end{matrix} & (12)\end{matrix}$

Now if one assumes that one has a likelihood function that factorizesinto N experiments, one has that one can take

$\begin{matrix}{\frac{\Delta}{P(E)} = {\frac{{\prod\limits_{j = 1}^{N}{P\left( {E_{j}\theta} \right)}} - {\prod\limits_{j = 1}^{N}{P^{\prime}\left( {E_{j}\theta} \right)}}}{\prod\limits_{j = 1}^{N}{P\left( {E_{j}\theta} \right)}} = {{\prod\limits_{j = 1}^{N}1} - {\prod\limits_{j = 1}^{N}{\frac{P^{\prime}\left( {E_{j}\theta} \right)}{P\left( {E_{j}\theta} \right)}.}}}}} & (13)\end{matrix}$

From the triangle inequality

$\begin{matrix}{{{\prod\limits_{j = 1}^{N}1} - {\prod\limits_{j = 1}^{N}\frac{P^{\prime}\left( {E_{j}\theta} \right)}{P\left( {E_{j}\theta} \right)}}} \leq {{{{\prod\limits_{j = 1}^{N - 1}1} - {\prod\limits_{j = 1}^{N - 1}\frac{P^{\prime}\left( {E_{j}\theta} \right)}{P\left( {E_{j}\theta} \right)}}}} + {{{1 - \frac{P^{\prime}\left( {E_{N}\theta} \right)}{P\left( {E_{N}\theta} \right)}}}\left( {1 + \gamma} \right)^{N}}} \leq {{{{\prod\limits_{j = 1}^{N - 1}1} - {\prod\limits_{j = 1}^{N - 1}\frac{P^{\prime}\left( {E_{j}\theta} \right)}{P\left( {E_{j}\theta} \right)}}}} + {{\gamma \left( {1 + \gamma} \right)}^{N}.}}} & (14)\end{matrix}$

Solving this recurrence relation gives

$\begin{matrix}{{{\prod\limits_{j = 1}^{N}1} - {\prod\limits_{j = 1}^{N}{{P^{\prime}\left( {E_{j}\theta} \right)}{P\left( {E_{j}\theta} \right)}}}} \leq {\left( {1 + \gamma} \right)^{N} - 1.}} & (15)\end{matrix}$

Thus the result follows.

V. Shift in the Posterior Mean from Using Random Hamiltonians

In this section, the shift in the posterior mean of the estimated phaseis analyzed assuming a random shift δ(ϕ) in the joint likelihood of allthe experiments,

P′({right arrow over (o)}|ϕ;{right arrow over (M)},{right arrow over(θ)})=P({right arrow over (o)}|ϕ;{right arrow over (M)},{right arrowover (θ)})+δ(ϕ).  (16)

Here, P({right arrow over (o)}|ϕ; {right arrow over (M)}, {right arrowover (θ)}) is the joint likelihood of a series of N outcomes {rightarrow over (o)} given a true phase ϕ and the experimental parameters{right arrow over (M)} and {right arrow over (θ)} for the originalHamiltonian. P′({right arrow over (o)}|ϕ; M, {right arrow over (θ)}) isthe joint likelihood with a new random Hamiltonian in each experiment.By a vector like {right arrow over (M)}, the repetitions are meant foreach experiment performed in the series; M_(i) is the number ofrepetitions in the i^(th) experiment.

First, one can work backwards from the assumption that the jointlikelihood is shifted by some amount δ(ϕ), to determine an upper boundon the acceptable difference in ground state energies between the trueand the random Hamiltonians. One can do this by working backwards fromthe shift in the joint likelihood of all experiments, to the shifts inthe likelihoods of individual experiments, and finally to thecorresponding tolerable differences between the ground state energies.Second, one can use this result to determine the shift in the posteriormean in terms of the differences in energies, as well as its standarddeviation over the ensemble of randomly generated Hamiltonians.

A. Shifts in the Joint Likelihood

The random Hamiltonians for each experiment lead to a random shift inthe joint likelihood of a series of outcomes

P′({right arrow over (o)}|ϕ;{right arrow over (M)},{right arrow over(θ)})=P({right arrow over (o)}|ϕ;{right arrow over (M)},{right arrowover (θ)})+δ(ϕ).  (17)

Assume that one would like to determine the maximum possible change inthe posterior mean under this shifted likelihood. One can work under theassumption that the mean shift in the likelihood over the prior is atmost |δ|≤P({right arrow over (o)})/2. The posterior is

$\begin{matrix}\begin{matrix}{{P^{\prime}\left( {{{\varphi \overset{\rightarrow}{o}};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)} = \frac{{P^{\prime}\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P(\varphi)}}{\int{{P^{\prime}\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P(\varphi)}d\; \varphi}}} \\{= \frac{{{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P(\varphi)}} + {{\delta (\varphi)}{P(\varphi)}}}{\int{\left( {{{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P(\varphi)}} + {{\delta (\varphi)}{P(\varphi)}}} \right)d\; \varphi}}} \\{= {\frac{{{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P(\theta)}} + {{\delta (\varphi)}{P(\varphi)}}}{{P\left( \overset{\rightarrow}{o} \right)} + \overset{\_}{\delta}}.}}\end{matrix} & (18)\end{matrix}$

One can make progress toward bounding the shift in the posterior byfirst bounding the shift in the joint likelihood in terms of the shiftsin the likelihoods of the individual experiments, as follows.Lemma 4. Let P(o_(j)|ϕ; M_(j), θ_(j)) be the likelihood of outcome o_(j)on the j^(th) experiment for the Hamiltonian H, and P′(o_(j)|ϕ; M_(j),θ_(j))=P(o_(j)|ϕ; M_(j), θ_(j))+ϵ_(j)(ϕ) be the likelihood with therandomly generated Hamiltonian H_(j). Assume that Nmax_(j)(|ϵ_(j)(ϕ)|/P(o_(j)|ϕ, M_(j), θ_(j)))<1 and |ϵ_(j)(ϕ)|≤P(o_(j)|ϕ,M_(j), θ_(j))/2 for all experiments j. Then the mean shift in the jointlikelihood of all N experiments,

|δ|=|∫P(ϕ)(P′({right arrow over (o)}|ϕ;{right arrow over (M)},{rightarrow over (θ)})−P({right arrow over (o)}|ϕ;{right arrow over(M)},{right arrow over (θ)}))dϕ|,

is at most

${\overset{\_}{\delta}} \leq {2{\sum\limits_{j = 1}^{N}{\max_{\varphi}{\frac{{\epsilon_{j}(\varphi)}}{P\left( {{{o_{j}\varphi};M_{j}},\theta_{j}} \right)}{{P\left( \overset{\rightarrow}{o} \right)}.}}}}}$

Proof. One can write the joint likelihood in terms of the shift ϵ_(j)(ϕ)to the likelihoods of each of the N experiments in the sequence,P′(o_(j)|ϕ; M_(j), θ_(j))=P(o_(j)|ϕ; M_(j), θ_(j))+ϵ_(j)(ϕ). The jointlikelihood is P′({right arrow over (o)}|ϕ; {right arrow over (M)},{right arrow over (θ)})=Π_(j=1) ^(N) (P(o_(j)|ϕ; M_(j),θ_(j))+ϵ_(j)(ϕ)), so

$\begin{matrix}\begin{matrix}{{\log \; {P^{\prime}\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}} = {\log \; \left( {\prod\limits_{j = 1}^{N}\left( {{P\left( {{{o_{j}\varphi};M_{j}},\theta_{j}} \right)} + {\epsilon_{j}(\varphi)}} \right)} \right)}} \\{= {{\sum\limits_{j = 1}^{N}{\log \; {P\left( {{o_{j}\varphi},M_{j},\theta_{j}} \right)}}} +}} \\{{\log \left( {1 + \frac{\epsilon_{j}(\varphi)}{P\left( {{o_{j}\varphi},M_{j},\theta_{j}} \right)}} \right)}} \\{= {{\log \; {P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}} +}} \\{{\sum\limits_{j = 1}^{N}{\log \left( {1 + \frac{\epsilon_{j}(\varphi)}{P\left( {{o_{j}\varphi},M_{j},\theta_{j}} \right)}} \right)}}}\end{matrix} & (19)\end{matrix}$

This gives one the ratio of the shifted to the unshifted jointlikelihood,

$\begin{matrix}{\frac{P^{\prime}\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)} = {{\exp \left\lbrack {\sum\limits_{j = 1}^{N}{\log \; \left( {1 + \frac{\epsilon_{j}(\varphi)}{P\left( {{o_{j}\varphi},M_{j},\theta_{j}} \right)}} \right)}} \right\rbrack}.}} & (20)\end{matrix}$

One can then, for example, linearize and simplify this usinginequalities for the logarithm and exponential. By finding inequalitieswhich either upper or lower bound both these functions, one can upperbound on |δ(ϕ)| in terms of the unshifted likelihoods P({right arrowover (o)}|ϕ; {right arrow over (M)}, {right arrow over (θ)}) andP(o_(j)|ϕ; M_(j), θ_(j)), and the shift in the single-experimentlikelihood ϵ_(j)(ϕ).

The inequalities that one can use to sandwich the ratio are

1 − |x| ≤ exp(x), for x ≤ 0 exp(x) ≤ 1 + 2|x|, for x <1 −2|x| ≤ log(1 +x), for |x| ≤ ½ log(1 + x) ≤ |x|, for x ϵ 

In order for all four inequalities to hold, one must have that Nmax_(j)(|ϵ_(j)(ϕ)|/P(o_(j)|ϕ, M_(j), θ_(j)))<1 (for the exponentialinequalities) and |ϵ_(j)(ϕ)|≤P(o_(j)|ϕ, M_(j), θ_(j))/2 for all j (forthe logarithm inequalities). Using them to upper bound the ratio of theshifted to the unshifted likelihood,

$\begin{matrix}{{\frac{P^{\prime}\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)} = {\frac{{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)} + {\delta (\varphi)}}{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)} \leq {\exp \left\lbrack {\sum\limits_{j = 1}^{N}\frac{{\epsilon_{j}(\varphi)}}{P\left( {{o_{j}\varphi},M_{j},\theta_{j}} \right)}} \right\rbrack} \leq {1 + {2{\sum\limits_{j = 1}^{N}\frac{{\epsilon_{j}(\varphi)}}{P\left( {{o_{j}\varphi},M_{j},\theta_{j}} \right)}}}}}};} & (22) \\{\mspace{79mu} {{\delta (\varphi)} \leq {2{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{\sum\limits_{j = 1}^{N}{\frac{{\epsilon_{j}(\varphi)}}{P\left( {{o_{j}\varphi},M_{j},\theta_{j}} \right)}.}}}}} & \;\end{matrix}$

On the other hand, using them to lower bound the ratio,

$\begin{matrix}{{\frac{P^{\prime}\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)} = {\frac{{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)} + {\delta (\varphi)}}{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)} \geq {\exp \left\lbrack {{- 2}{\sum\limits_{j = 1}^{N}\frac{{\epsilon_{j}(\varphi)}}{P\left( {{{o_{j}\varphi};M_{j}},\theta_{j}} \right)}}} \right\rbrack} \geq {1 - {2{\sum\limits_{j = 1}^{N}\frac{{\epsilon_{j}(\varphi)}}{P\left( {{{o_{j}\varphi};M_{j}},\theta_{j}} \right)}}}}}};} & (23) \\{\mspace{79mu} {{\delta (\varphi)} \geq {{- 2}{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{\sum\limits_{j = 1}^{N}{\frac{{\epsilon_{j}(\varphi)}}{P\left( {{{o_{j}\varphi};M_{j}},\theta_{j}} \right)}.}}}}} & \;\end{matrix}$

The upper and lower bounds are identical up to sign. This allows one tocombine them directly, so one has

$\begin{matrix}{{{\delta (\varphi)}} \leq {2{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{\sum\limits_{j = 1}^{N}{\frac{{\epsilon_{j}(\varphi)}}{P\left( {{{o_{j}\varphi};M_{j}},\theta_{j}} \right)}.}}}} & (24)\end{matrix}$

From this, one can find an upper bound on the mean shift over theposterior, |δ|, since by the triangle inequality

$\begin{matrix}{{\overset{\_}{\delta}} = {{{\int{{\delta (\varphi)}{P(\varphi)}d\; \varphi}}} \leq {2{\sum\limits_{j = 1}^{N}{\int{\left( {\frac{{\epsilon_{j}(\varphi)}}{P\left( {{{o_{j}\varphi};M_{j}},\theta_{j}} \right)}{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P(\varphi)}} \right)d\; \varphi}}}} \leq {2{\sum\limits_{j = 1}^{N}{\max\limits_{\varphi}{\frac{{\epsilon_{j}(\varphi)}}{P\left( {{{o_{j}\varphi};M_{j}},\theta_{j}} \right)}{{P\left( \overset{\rightarrow}{o} \right)}.}}}}}}} & (25)\end{matrix}$

So one has a bound on the shift in the joint likelihood in terms of theshifts in the likelihoods of individual experiments. These results allowone to bound the shift in the posterior mean in terms of the shifts inthe likelihoods of the individual experiments ϵ_(j)(ϕ).

B. Shift in the Posterior Mean

One can use the assumption that |δ|≤P({right arrow over (o)})/2 to boundthe shift in the posterior mean.

Lemma 5. Assuming in addition to the assumptions of Lemma 4 that|δ|≤P({right arrow over (o)})/2, the difference between the theposterior mean that one would see with the ideal likelihood function andthe perturbed likelihood function is at most

${{\overset{\_}{\varphi} - {\overset{\_}{\varphi}}^{\prime}}} \leq {8\; {\max\limits_{\varphi}\; {\left( {\sum\limits_{j = 1}^{N}\frac{{\epsilon_{j}(\varphi)}}{P\left( {{o_{j}\varphi},M_{j},\theta_{j}} \right)}} \right)\mspace{11mu} {{\overset{\_}{\varphi}}^{post}.}}}}$

Proof. One can approach the problem of bounding the difference betweenthe posterior means by bounding the point-wise difference between theshifted posterior and the posterior with the original Hamiltonian,

$\begin{matrix}{{{{P\left( {{{\varphi \overset{\rightarrow}{o}};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)} - {P^{\prime}\left( {{{\varphi \overset{\rightarrow}{o}};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}}} = {{{\frac{{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P(\varphi)}}{P\left( \overset{\rightarrow}{o} \right)} - \frac{{{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P(\varphi)}} + {{\delta (\varphi)}{P(\varphi)}}}{{P\left( \overset{\rightarrow}{o} \right)} + \overset{\_}{\delta}}}}.}} & (26)\end{matrix}$

As a first step, one can place an upper bound on the denominator of theshifted posterior,

$\begin{matrix}{{\left( {{P\left( \overset{\rightarrow}{o} \right)} + \overset{\_}{\delta}} \right)\text{:}}\begin{matrix}{\frac{1}{{P\left( \overset{\rightarrow}{o} \right)} + \overset{\_}{\delta}} = {\frac{1}{P\left( \overset{\rightarrow}{o} \right)}{\sum\limits_{k = 0}^{\infty}\left( \frac{- \overset{\_}{\delta}}{P\left( \overset{\rightarrow}{o} \right)} \right)^{k}}}} \\{= {\frac{1}{P\left( \overset{\rightarrow}{o} \right)} - \frac{\overset{\_}{\delta}}{{P\left( \overset{\rightarrow}{o} \right)}^{2}} + {\frac{{\overset{\_}{\delta}}^{2}}{{P\left( \overset{\rightarrow}{o} \right)}^{3}}{\sum\limits_{k = 0}^{\infty}\left( \frac{- \overset{\_}{\delta}}{P\left( \overset{\rightarrow}{o} \right)} \right)^{k}}}}} \\{{{\leq {\frac{1}{P\left( \overset{\rightarrow}{o} \right)} + \frac{2{\overset{\_}{\delta}}}{{P\left( \overset{\rightarrow}{o} \right)}^{2}}}} = \frac{1 + {2{{\overset{\_}{\delta}}/{P\left( \overset{\rightarrow}{o} \right)}}}}{P\left( \overset{\rightarrow}{o} \right)}},}\end{matrix}} & (27)\end{matrix}$

where in the two inequalities the assumption that |δ|≤P({right arrowover (o)})/2 was used. Using this, the point-wise difference between theposteriors is at most

$\begin{matrix}{{{{\frac{{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P(\varphi)}}{P\left( \overset{\rightarrow}{o} \right)} - \frac{{{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P(\varphi)}} + {{\delta (\varphi)}{P(\varphi)}}}{{P\left( \overset{\rightarrow}{o} \right)} + \overset{\_}{\delta}}}} \leq {{{\frac{{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P(\varphi)}}{P\left( \overset{\rightarrow}{o} \right)} - \frac{{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P(\varphi)}}{{P\left( \overset{\rightarrow}{o} \right)} + \overset{\_}{\delta}}}} + {\frac{{\delta (\varphi)}{P(\varphi)}}{{P\left( \overset{\rightarrow}{o} \right)} + \overset{\_}{\delta}}}} \leq {\frac{2{\overset{\_}{\delta}}{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P(\varphi)}}{{P\left( \overset{\rightarrow}{o} \right)}^{2}} + {\frac{{{\delta (\varphi)}}{P(\varphi)}}{P\left( \overset{\rightarrow}{o} \right)}\left( {1 + \frac{2{\overset{\_}{\delta}}}{P\left( \overset{\rightarrow}{o} \right)}} \right)}} \leq {\frac{2{\overset{\_}{\delta}}{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P(\varphi)}}{{P\left( \overset{\rightarrow}{o} \right)}^{2}} + \frac{2{{\delta (\varphi)}}{P(\varphi)}}{P\left( \overset{\rightarrow}{o} \right)}}},} & (28)\end{matrix}$

again using |δ|≤P({right arrow over (o)})/2. With this, one can boundthe change in the posterior mean,

$\begin{matrix}\begin{matrix}{{{{\overset{\_}{\varphi} - {\overset{\_}{\varphi}}^{\prime}}} \leq {\int{\varphi }}}{{P\left( {{{\varphi \overset{\rightarrow}{o}};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)} - {{P^{\prime}\left( {\varphi \left. {{\overset{\rightarrow}{o};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)} \right.}d\; \varphi}}} \\{\leq {\frac{2}{P\left( \overset{\rightarrow}{o} \right)}{\int{{\varphi }\left( {\frac{{\overset{\_}{\delta}}{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P(\varphi)}}{P\left( \overset{\rightarrow}{o} \right)} + {{{\delta (\varphi)}}{P(\varphi)}}} \right)\mspace{2mu} d\; \varphi}}}} \\{\leq {{\frac{2}{P\left( \overset{\rightarrow}{o} \right)}{\int{{\varphi }\; {{\delta (\varphi)}}{P(\varphi)}d\; \varphi}}} + {\frac{2{\overset{\_}{\delta}}}{P\left( \overset{\rightarrow}{o} \right)}{\int{{\varphi }\left( \frac{{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P(\varphi)}}{P\left( \overset{\rightarrow}{o} \right)} \right)\; d\; \varphi}}}}} \\{\leq {\frac{2}{P\left( \overset{\rightarrow}{o} \right)}\left( {{\int{{\varphi }\; {{\delta (\varphi)}}{P(\varphi)}d\; \varphi}} + {{\overset{\_}{\varphi}}^{post}{\overset{\_}{\delta}}}} \right)}}\end{matrix} & (29)\end{matrix}$

Now, the bounds from Lemma 4 allow one to bound the shift on theposterior mean in terms of the shifts in the likelihoods of individualexperiments,

(ϕ),

$\begin{matrix}{{{{\overset{\_}{\varphi} - {\overset{\_}{\varphi}}^{\prime}}} \leq {\frac{2}{P\left( \overset{\rightarrow}{o} \right)}\left( {{\int{{\varphi }\; {{\delta (\varphi)}}{P(\varphi)}d\; \varphi}} + {{\overset{\_}{\varphi}}^{post}{\overset{\_}{\delta}}}} \right)} \leq {\frac{2}{P\left( \overset{\rightarrow}{o} \right)}\left( {{2\; {\max\limits_{\varphi}{\sum\limits_{j = 1}^{N}{\frac{{\epsilon_{j}(\varphi)}}{P\left( {{o_{j}\varphi},M_{j},\theta_{j}} \right)}{P\left( \overset{\rightarrow}{o} \right)}{\int{{\varphi }\left( \frac{{P\left( {{{\overset{\rightarrow}{o}\varphi};\overset{\rightarrow}{M}},\overset{\rightarrow}{\theta}} \right)}{P(\varphi)}}{P\left( \overset{\rightarrow}{o} \right)} \right)\; d\; \varphi}}}}}} + {{\overset{\_}{\varphi}}^{post}{\overset{\_}{\delta}}}} \right)}},} & (30)\end{matrix}$

where in the last step, one can multiple and divide by P({right arrowover (o)}). This is

$\begin{matrix}\begin{matrix}{{{\overset{\_}{\varphi} - {\overset{\_}{\varphi}}^{\prime}}} \leq {\frac{2}{P\left( \overset{\rightarrow}{o} \right)}\left( {{2\; {\max\limits_{\varphi}{\left( {\sum\limits_{j = 1}^{N}\frac{{\epsilon_{j}(\varphi)}}{P\left( {{o_{j}\varphi},M_{j},\theta_{j}} \right)}} \right){P\left( \overset{\rightarrow}{o} \right)}{\overset{\_}{\varphi}}^{post}}}} + {{\overset{\_}{\varphi}}^{post}{\overset{\_}{\delta}}}} \right)}} \\{\leq {8\; {\max\limits_{\varphi}{\left( {\sum\limits_{j = 1}^{N}\frac{{\epsilon_{j}(\varphi)}}{P\left( {{o_{j}\varphi},M_{j},\theta_{j}} \right)}} \right){{\overset{\_}{\varphi}}^{post}.}}}}}\end{matrix} & (31)\end{matrix}$

C. Acceptable Shifts in the Phase

A further question is what the bound on the shift in the posterior meanis in terms of shifts in the phase.

Theorem 6. If the assumptions of Lemma 5, for all j and x

${{\in {\left\lbrack {{- \pi},\pi} \right){P\left( {{\left. o_{j} \middle| \theta \right.;x},\theta_{j}} \right)}}} = \frac{1 + {\left( {- 1} \right)^{o_{j}}{\cos \left( {M_{j}\left( {\theta_{j} - x} \right)} \right)}}}{2}},$

for each of the N experiments, one has that the eigenphases used in PE{ϕ′_(j)=1, . . . N} and the eigenphase the true Hamiltonian ϕ obey|ϕ−ϕ′_(j)≤Δϕ and additionally P(o_(j)|ϕ, M_(j), θ_(j))ϵΘ(1) then one hasthat the shift in the posterior mean of the eigenphase that arises frominaccuracies in the eigenvalues in the intervening Hamiltonians obeys

${{\overset{\_}{\varphi} - {\overset{\_}{\varphi}}^{\prime}}} \leq {8\; {\max\limits_{\varphi}{\left( {\sum\limits_{j = 1}^{N}\frac{M_{j}}{P\left( {{{o_{j}\varphi};M_{j}},\theta_{j}} \right)}} \right)\mspace{11mu} {{{\Delta \; \varphi}}.}}}}$

Furthermore if Σ_(j) M_(j)ϵO(1/ϵ_(ϕ)) and P(o_(j)|ϕ; M_(j), θ_(j))ϵΘ(1)for all j then

${{\overset{\_}{\varphi} - {\overset{\_}{\varphi}}^{\prime}}} \in {O\left( \frac{{\Delta \; \varphi}}{\epsilon_{\varphi}} \right)}$

Proof. One can express the shift in the posterior mean in terms of theshift in the phase applied to the ground state, Δϕ, by bounding ϵ_(j)(ϕ)in terms of it. Recall that the likelihood with the random Hamiltonianis

P′(o _(j) |ϕ;M _(j),θ_(j))=P(o _(j) |ϕ;M _(j),θ_(j))+ϵ_(j)(ϕ),  (32)

where the unshifted likelihood for the j^(th) experiment is P(o_(j)|ϕ;M_(j), θ_(j))=½(1+(−1)^(o) ^(j) cos(M_(j)(ϕ−θ_(j))). Thus,

|ϵ_(j)(ϕ)|=½|cos(M _(j)(ϕ+Δϕ−θ_(j))−cos(M _(j)(ϕ−θ_(j))|≤M_(j)|Δϕ|,  (33)

using the upper bound on the derivative sin(x)≤|x|. In sum, one has thatthe error in the posterior mean in the posterior mean is at most

$\begin{matrix}{{{\overset{\_}{\varphi} - {\overset{\_}{\varphi}}^{\prime}}} \leq {8\; {\max\limits_{\varphi}{\left( {\sum\limits_{j = 1}^{N}\frac{M_{j}}{P\left( {\left. o_{j} \middle| \varphi \right.,M_{j},\theta_{j}} \right)}} \right){\overset{\_}{\varphi }}^{post}{{{\Delta\varphi}}.}}}}} & (34)\end{matrix}$

The result then follows from the fact that the absolute value of theposterior mean is at most π if the branch [−π, π) is chosen.

VI. Shift in the Eigenphase with a New Random Hamiltonian in EachRepetition

One can reduce the variance in the applied phase by generating adifferent Hamiltonian in each repetition. However, this is not withoutits costs: it can be viewed either as leading to a failure probabilityin the evolution or more generally to an additional phase shift.

The reason this reduces the variance is somewhat complex to formalizemathematically—it comes in when you compute the variance of |Δϕ|.Instead of just having the indices for a single Hamiltonian, thevariance is over the indices of M_(j) Hamiltonians. Because of this, itonly scales as M

οϕ_(est)] instead of M²

[ϕ_(est)] as it usually would (there is an underlying variance inϕ_(est)). The cost is that, by reducing the variance in the phase inthis way, one causes an additional shift in the phase. If one did not dothis across multiple steps, one would be adiabatic all the way with thesame wrong Hamiltonian. Instead, this means that one is onlyapproximately adiabatic at the cost of the variance being lower by afactor M_(j). Since the additional shift is also linear in [M_(j)], thiscan lead to an improvement. It generally requires that the gap be small,and that λ_(j)∝∥H_(j)−H_(j-1)∥ be small.

One then has a competition between the standard deviation being M_(j)

[ϕ_(est)] or √{square root over (M_(j))}

[ϕ_(est)], and this new shift which is linear in M_(j). So depending onthe gap, it might be hard to get a rigorous bound showing that this isbetter, and that one should not just stick with the higher variance froma single Hamiltonian.

A. Failure Probability of the Algorithm

For phase estimation, one can reduce the variance of the estimate in thephase by randomizing within the repetitions for each experiment. For thej^(th) experiment with M_(j) repetitions (recall that M_(j) is notnecessarily an integer), one divides into [M_(j)] repetitions.

Within each repetition, one can randomly generate a new HamiltonianH_(k). Each Hamiltonian H_(k) has a slightly different ground state andenergy than all the others.

The reason this reduces the variance in the estimated phase is that thephases between repetitions are uncorrelated whereas for thesingle-Hamiltonian case, the variance in the phase exp(−iMϕ_(est)) is

[Mϕ_(est)]=M²

[ϕ_(est)], when one simulates a different random Hamiltonian in eachrepetition (and estimate the sum of the phases, as exp(−iΣ_(k=1) ^(M)ϕ_(k,est))), the variance is

[Σ_(k=1) ^(M) ϕ_(k,est)]=Σ_(k=1) ^(M)

[ϕ_(k,est)].

By evolving under a different random instantiation of the Hamiltonian ineach repetition, the variance in the phase is quadratically reduced; theonly cost is that the algorithm now has either a failure probability (ofleaving the ground state from repetition to repetition, e.g. in thetransition from the ground state of H_(k−1) to the ground state ofH_(k)) or an additional phase shift compared to the true sum of theground state energies. The first case is simpler to analyze: it is shownin Lemma 7, provided that the gap is sufficiently small, that thefailure probability can be made arbitrarily small. One can do this byviewing the success probability of the algorithm as the probability ofremaining in the ground state throughout the sequence of [M_(j)] randomHamiltonians. In the second case, one can prove in Lemma 8 a bound onthe difference between eigenvalues if the state only leaves the groundspace for short intervals during the evolution.

Lemma 7. Consider a sequence of Hamiltonians {H_(k)}_(k=1) ^(M), M>1.Let γ be the minimum gap between the ground and first excited energiesof any of the Hamiltonians, γ=min_(k)(E₁ ^(k)−E₀ ^(k)). Similarly, letλ=max_(k)∥H_(k)−H_(k−1)∥ be the maximum difference between any two inthe sequence. The probability of leaving the ground state whentransferring from H_(i) to H₂ through to H_(M) in order is at most 0<ϵ<1provided that

$\frac{\lambda}{\gamma} < {\sqrt{1 - {\exp \left( \frac{\log \left( {1 - c} \right)}{M - 1} \right)}}.}$

Proof. Let |ψ_(i) ^(k)

be the i^(th) eigenstate of the Hamiltonian H_(k) and let E_(i) ^(k) bethe corresponding energy. Given that the algorithm begins in the groundstate of H₁ (|ψ₀ ¹

), the probability of remaining in the ground state through all M stepsis

|

ψ₀ ^(M)|ψ₀ ^(M-1)

. . .

ψ₀ ²|ψ₀ ¹

|².  (35)

This is the probability of the algorithm staying in the ground state inevery segment. One can simplify this expression by finding a bound for∥ψ₀ ^(k)

−|ψ₀ ^(k−1)

|². Let λ_(k)V_(k)=H_(k)−H_(k−1), where one can choose λ_(k) such that∥V_(k)∥=1 to simplify the proof. Treating λ_(k)V_(k) as a perturbationon H_(k−1), the components of the shift in the ground state of H_(k−1)are bounded by the derivative

$\begin{matrix}{{{\frac{\partial}{\partial\lambda}{\langle\left. \psi_{0}^{k - 1} \middle| \psi_{}^{k} \right.\rangle}}} = \frac{{\langle{\psi_{}^{k - 1}{V_{k}}\psi_{0}^{k - 1}}\rangle}}{E_{}^{k - 1} - E_{0}^{k - 1}}} & (36)\end{matrix}$

multiplied by λ=max |λ_(k)|, where the maximization is over both k aswell as perturbations for a given k. Using this,

$\begin{matrix}{{{\langle\left. \psi_{}^{k} \middle| \psi_{0}^{k - 1} \right.\rangle}}^{2} \leq {\lambda^{2}\frac{{{\langle{\psi_{}^{k - 1}{V_{k}}\psi_{0}^{k - 1}}\rangle}}^{2}}{\left( {E_{}^{k - 1} - E_{0}^{k - 1}} \right)^{2}}} \leq {\lambda^{2}{\frac{{{\langle{\psi_{}^{k - 1}{V_{k}}\psi_{0}^{k - 1}}\rangle}}^{2}}{\gamma^{2}}.}}} & (37)\end{matrix}$

This allows one to write |ψ₀ ^(k+1)

=(1+δ₀)|ψ₀ ^(k)

+

|

, where |

≤λmax_(k)

$\frac{{\langle\left. \psi_{}^{k} \middle| V_{k} \middle| \psi_{0}^{k} \right.\rangle}}{E_{0}^{k} - E_{}^{k}}.$

Letting V_(k)|ψ₀ ^(k)

=κ_(k)|ϕ_(k)), where one can again choose κ_(k) such that |ϕ_(k)

is normalized,

$\begin{matrix}\begin{matrix}{{{{\psi_{0}^{k}\rangle} - {\psi_{0}^{k - 1}\rangle}}}^{2} = {{\delta_{0}^{2} + {\sum\limits_{ > 0}\delta_{}^{2}}} \leq {\delta_{0}^{2} + {\frac{\lambda^{2}}{\gamma^{2}}{\sum\limits_{}{{\langle{\psi_{}^{k - 1}{V_{k}}\psi_{0}^{k - 1}}\rangle}}^{2}}}}}} \\{= {\delta_{0}^{2} + {\frac{\lambda^{2}}{\gamma^{2}}\kappa_{k}^{2}{\sum\limits_{}{{\langle\left. \psi_{}^{k} \middle| \varphi_{k} \right.\rangle}}^{2}}}}} \\{= {\delta_{0}^{2} + {\frac{\lambda^{2}}{\gamma^{2}}{\kappa_{k}^{2}.}}}}\end{matrix} & (38)\end{matrix}$

One can solve for δ₀ ² in terms of

since (1+δ₀)²+

=1. Since √{square root over (1−x)}≥1−x for xϵ[0, 1],

$\begin{matrix}{\delta_{0}^{2} = {\left( {1 - \sqrt{1 - {\sum\limits_{ > 0}\delta_{}^{2}}}} \right)^{2} \leq \left( {\sum\limits_{ > 0}\delta_{}^{2}} \right)^{2} \leq {\sum\limits_{ > 0}\delta_{}^{2}}}} & (39)\end{matrix}$

since

. Finally, returning to ∥ψ₀ ^(k)

−|ψ₀ ^(k−1)

|², since κ_(k)≤1 (this is true because ∥V_(k)∥=1), the differencebetween the ground states of the two Hamiltonians is at most

$\begin{matrix}{{{{\psi_{0}^{k}\rangle} - {\psi_{0}^{k - 1}\rangle}}}^{2} \leq \frac{2\lambda^{2}}{\gamma^{2}}} & (40)\end{matrix}$

This means that the overlap probability between the ground states of anytwo adjacent Hamiltonians is

${{\langle\left. \psi_{0}^{k + 1} \middle| \psi_{0}^{k} \right.\rangle}}^{2} \geq {1 - {\frac{\lambda^{2}}{\gamma^{2}}.}}$

Across M segments (M−1 transitions), the success probability is at least

$\left( {1 - \frac{\lambda^{2}}{\gamma^{2}}} \right)^{M - 1}.$

If one wishes for the failure probability to be at most some fixed0<ϵ<1, one must have

$\begin{matrix}{{\left( {1 - \frac{\lambda^{2}}{\gamma^{2}}} \right)^{M - 1} > {1 - \epsilon}}{\frac{\lambda}{\gamma} < {\sqrt{1 - {\exp \left( \frac{\log \left( {1 - \epsilon} \right)}{M - 1} \right)}}.}}} & (41)\end{matrix}$

If one can only prepare the ground state |ψ₀

of the original Hamiltonian, the success probability has an additionalfactor |

ψ₀ ¹|ψ₀

|². In this case, one can apply Lemma 7 with ∥H−H₁∥ included in themaximization for λ. Further, since γ=min_(k)(E₁ ^(k)−E₀ ^(k))≤E₁−E₀−2λ|

ψ₀|H_(k)−H|ψ₀

≤E₁−E₀−2λ, where E₁−E₀ is the gap between the ground and first excitedstates of H, one needs

$\begin{matrix}{\frac{\lambda}{E_{1} - E_{0} - {2\lambda}} < {\sqrt{1 - {\exp \left( \frac{\log \left( {1 - \epsilon} \right)}{M} \right)}}.}} & (42)\end{matrix}$

Provided that this occurs, one stays in the ground state of eachHamiltonian throughout the simulation with probability 1−ϵ. In thiscase, the total accumulated phase is

$\begin{matrix}{{\frac{\begin{matrix}{{\langle\psi_{0}^{\lceil M_{j}\rceil}}e^{{- {iH}_{\lceil M_{j}\rceil}}\Delta \; t}{\psi_{0}^{{\lceil M_{j}\rceil} - 1}\rangle}\mspace{14mu} \ldots \mspace{14mu} {\langle\psi_{0}^{2}}e^{{- {iH}_{2}}\Delta \; t}{\psi_{0}^{1}\rangle}} \\{{\langle\psi_{0}^{1}}e^{{- {iH}_{1}}\Delta \; t}{\psi_{0}\rangle}}\end{matrix}}{{\langle\left. \psi_{0}^{\lceil M_{j}\rceil} \middle| \psi_{0}^{{\lceil M_{j}\rceil} - 1} \right.\rangle}\ldots {\langle\left. \psi_{0}^{2} \middle| \psi_{0}^{1} \right.\rangle}{\langle\left. \psi_{0}^{1} \middle| \psi_{0} \right.\rangle}} = {\exp \left( {{- i}{\sum\limits_{k = 1}^{\lceil{Mj}\rceil}{E_{k}^{0}\Delta \; t}}} \right)}},\mspace{20mu} {{{where}\mspace{14mu} \Delta \; t} = {M_{j}{t/{\left\lceil M_{j} \right\rceil.}}}}} & (43)\end{matrix}$

B. Phase Shifts Due to Hamiltonian Errors

One can generalize the analysis of the difference in the phase bydetermining the difference between the desired (adiabatic) unitary andthe true one. Evolving under M random Hamiltonians in sequence, theunitary applied for each new Hamiltonian H_(k) is

$\begin{matrix}{{U_{k} = {{\exp \left( {{- {iH}_{k}}\Delta \; t} \right)} = {\sum\limits_{}{{\psi_{}^{k}\rangle}{\langle\psi_{}^{k}}e^{{- {iE}_{}^{k}}\Delta \; t}}}}},} & (44)\end{matrix}$

while the adiabatic unitary one would ideally apply is

$\begin{matrix}{U_{k,{ad}} = {\sum\limits_{}{{\psi_{}^{k + 1}\rangle}{\langle\psi_{}^{k}}{e^{{- {iE}_{}^{k}}\Delta \; t}.}}}} & (45)\end{matrix}$

The difference between the two is that true time evolution U_(k) underH_(k) applies phases to the eigenstates of H_(k), while the adiabaticunitary U_(k,ad) applies the eigenphase, and then maps each eigenstateof H_(k) to the corresponding eigenstate of H_(k+1). This means that ifthe system begins in the ground state of H₁, the phase which will beapplied to it by the sequence (U_([M) _(j) _(],ad)U_([M) _(j) _(]−1,ad). . . U_(2,ad)U_(1,ad) is proportional to the sum of the ground stateenergies of each Hamiltonian in that sequence. By comparison, U_([M)_(j) _(])U_([M) _(j) _(]−1) . . . U₂U₁ will include contributions frommany different eigenstates of the different Hamiltonians H_(k).

One can bound the difference between the unitaries U_(k) and U_(k,ad) asfollows.

Lemma 8. Let P₀ ^(k) be the projector onto the ground state of H_(k),|ψ₀ ^(k)

, and let the assumptions of Lemma 7 hold. The difference between theeigenvalues of

U_(k)P₀^(k) = exp (−iH_(k)Δ t)P₀^(k) = ∑_()ψ_()^(k)⟩⟨ψ_()^(k)e^(−iE_()^(k)Δ t)P₀^(k)  andU_(k, ad)P₀^(k) = ψ₀^(k + 1)⟩⟨ψ₀^(k)e^(−iE_()^(k)Δ t)P₀^(k),

where Δt is the simulation time, is at most

${{\left( {U_{k} - U_{k,{ad}}} \right)P_{0}^{k}}} \leq {\frac{2\; \lambda^{2}}{\left( {\gamma - {2\; \lambda}} \right)^{2}}.}$

Proof. First, one can expand the true unitary using the resolution ofthe identity Σ_(p)|ψ_(p) ^(k+1)

ψ_(p) ^(k+1)|, the eigenstates of the next Hamiltonian, H_(k+1):

$\begin{matrix}{U_{k} = {\sum\limits_{p,}{{\psi_{p}^{k + 1}\rangle}{\langle{\psi_{p}^{k + 1}{\psi_{}^{k}\rangle}{\langle\psi_{}^{k}}{e^{{- {iE}_{}^{k}}\Delta \; t}.}}}}}} & (46)\end{matrix}$

Let

=

ψ_(p) ^(k+1)|ψ_(l) ^(k)

for p≠

and 1+Δ_(pp)=

_(p) ^(k+1)|ψ_(p) ^(k)

when p=

. In a sense, one is writing the new eigenstate |ψ_(p) ^(k+1)

as a slight shift from the state |

): this e reason that one chooses

ψ_(p) ^(k+1)|ψ_(p) ^(k)

=1+Δ_(pp). Using this definition, one can continue to simplify U_(k), as

$\begin{matrix}{U_{k} = {{\sum\limits_{p}{\left( {1 + \Delta_{pp}} \right){\psi_{p}^{k + 1}\rangle}{\langle\psi_{p}^{k}}e^{{- {iE}_{p}^{k}}\Delta \; t}}} + {\sum\limits_{p \neq }{{\Delta \;}_{p\; }{\psi_{p}^{k + 1}\rangle}{\langle\psi_{}^{k}}{e^{{- {iE}_{}^{k}}\Delta \; t}.}}}}} & (47)\end{matrix}$

One is now well-positioned to bound ∥(U_(k)−U_(k,ad))P₀ ^(k)∥. Notingthat U_(k,ad) exactly equals the 1 in the first sum in U_(k).

$\begin{matrix}{{{\left( {U_{k} - U_{k,{ad}}} \right)P_{0}^{k}}} = {{{\sum\limits_{p}{{\Delta \;}_{p\; }{\psi_{p}^{k + 1}\rangle}{\langle\psi_{}^{k}}e^{{- {iE}_{}^{k}}\Delta \; t}{\psi_{0}^{k}\rangle}{\langle\psi_{0}^{k}}}}} = {{\max\limits_{\psi\rangle}{{\sum\limits_{p\; }{{\Delta \;}_{p\; }{\psi_{p}^{k + 1}\rangle}{\langle\psi_{}^{k}}e^{{- {iE}_{}^{k}}\Delta \; t}{\psi_{0}^{k}\rangle}{\langle\psi_{0}^{k}}\psi}}}^{2}} = {{{\sum\limits_{p}{\Delta_{p\; 0}e^{{- {iE}_{0}^{k}}\Delta \; t}}}}^{2} \leq {\sum\limits_{p}{{\Delta_{p\; 0}}^{2}.}}}}}} & (48)\end{matrix}$

The final step in bounding ∥U_(k)−U_(k,ad)∥ is to bound |

|=

ψ_(p) ^(k+1)|

similarly to how one bounded

. For p≠

,

is given by

$\begin{matrix}{{{{\Delta_{p\; }}^{2} = {{{\langle\psi_{p}^{k + 1}}\psi_{}^{k}}\rangle}}}^{2} \leq {\lambda^{2}{\frac{{{{\langle\psi_{p}^{k}}V_{k}{\psi_{}^{k}\rangle}}}^{2}}{\left( {E_{p}^{k} - E_{}^{k}} \right)^{2}}.}}} & (49)\end{matrix}$

So, as with the bounds on |δ₀|² and Σ_(l>0)|δ_(l)|² in Lemma 7,∥(U_(k)−U_(k,ad))P₀ ^(k)∥ is upper bounded by

$\begin{matrix}{{\sum\limits_{p}{\Delta_{p\; 0}}^{2}} = {\sum\limits_{p}{{{{\langle{\psi_{p}^{k + 1}{\psi_{0}^{k}\rangle}}}^{2} \leq {\sum\limits_{p}{\lambda^{2}\frac{{{{\langle\psi_{p}^{k}}V_{k}{\psi_{0}^{k}\rangle}}}^{2}}{\left( {E_{p}^{k} - E_{0}^{k}} \right)^{2}}}} \leq \frac{{2\lambda^{2}}\;}{\left( {\gamma - {2\; \lambda}} \right)^{2}}},}}}} & (50)\end{matrix}$

which completes the proof.Theorem 9. Consider a sequence of Hamiltonians {H_(k)}_(k=1) ^(M), M>1.Let γ be the minimum gap between the ground and first excited energiesof any of the Hamiltonians, γ=min_(k)(E₁ ^(k)−E₀ ^(k)). Similarly, letλ=max_(k)∥H_(k)−H_(k−1)∥ be the maximum difference between any twoHamiltonians in the sequence. The maximum error in the estimatedeigenphases of the unitary found by the products of these M Hamiltoniansis at most

${{{\varphi_{est} - \varphi_{true}}} \leq \frac{{2\; M\; \lambda^{2}}\;}{\left( {\gamma - {2\; \lambda}} \right)^{2}}},$

with a probability of failure of at most e provided that

$\frac{\lambda}{\gamma} < {\sqrt{1 - {\exp \left( \frac{\log \left( {1 - \epsilon} \right)}{M - 1} \right)}}.}$

Proof. Lemma 8 gives the difference between eigenvalues of U_(k)P₀ ^(k)and U_(k,ad)P₀ ^(k). Across the entire sequence, one has

$\begin{matrix}{{{{U_{M}P_{0}^{M}\mspace{14mu} \ldots \mspace{14mu} U_{k}P_{0}^{k}\mspace{14mu} \ldots \mspace{14mu} U_{1}P_{0}^{1}} - {U_{M,{ad}}P_{0}^{M}\mspace{14mu} \ldots \mspace{14mu} U_{k,{ad}}P_{0}^{k}\mspace{25mu} \ldots \mspace{11mu} U_{1,{ad}}P_{0}^{1}}}}\; \leq {\frac{{2\; M\; \lambda^{2}}\;}{\left( {\gamma - {2\; \lambda}} \right)^{2}}.}} & (51)\end{matrix}$

This is the maximum possible difference between the accumulated phasesfor the ideal and actual sequences, assuming the system leaves theground state for at most one repetition at a time.

The probability of leaving the groundstate as part of a Landau-Zenerprocess instigated by the measurement at adjacent values of theHamiltonians is, under the assumptions of Lemma 7, that the failureprobability occurring at each projection is ϵ if

$\begin{matrix}{{\frac{\lambda}{\gamma} < \sqrt{1 - {\exp \left( \frac{\log \left( {1 - \epsilon} \right)}{M - 1} \right)}}},} & (52)\end{matrix}$

thus the result follows trivially from these two results.

VII. Importance Sampling

The fundamental idea behind the example approach to decimation of aHamiltonian is importance sampling. This approach has already seen greatuse in coalescing, but one can use it slightly differently here. Theidea behind importance sampling is to reduce the variance of the mean ofa quantity by reweighting the sum. Specifically, one can write the meanof N numbers F(j) as

$\begin{matrix}{{{\frac{1}{N}{\sum\limits_{j}{F(j)}}} = {\sum\limits_{j}{{f(j)}\frac{F(j)}{{Nf}(j)}}}},} & (53)\end{matrix}$

where ƒ(j) is the importance of a given term. This shows that one canview the initial unweighted average as the average of a reweightedquantity x_(j)/(ƒ(j)N). While this does not have an impact on the meanof x_(j) it can dramatically reduce the sample variance of the mean andthus is widely used in statistics to provide more accurate estimates ofmeans. The optimal importance function to take in these cases isƒ(j)∝|x_(j)|. In such cases, it is straightforward to see that thevariance of the resulting distribution is in fact zero if the sign ofthe x_(j) is constant. In such cases a straight forward calculationshows that this optimal variance is

_(ƒ) _(opt) =(

(|F|))²−(

(F))²  (54)

The optimal variance in (54) is in fact zero if the sign of the numbersis constant. While this may seem surprising, it becomes less mysteriouswhen one notes that in order to compute the optimal importance functionyou need the ensemble mean that you would like to estimate. This woulddefeat the purpose of importance sampling in most cases. Thus if onewants to glean an advantage from importance sampling for Hamiltoniansimulation then it is important to show that one can use it even with aninexact importance function that can be, for example, computedefficiently using a classical computer.

It is now shown how this robustness holds below.

Lemma 10. Let F:

_(N)

with

(F)=N⁻¹Σ_(j=0) ^(N-1) F(j) be an unknown probability distribution thatcan be sampled from and let {tilde over (F)}:

_(N)

be a known function such that for all j, |{tilde over(F)}(j)|−|F(j)|=δ_(j) with |δ_(j)|≤|F(j)|/2. If importance sampling isused with an importance function ƒ(j)=|{tilde over (F)}(j)|/Σ_(k)|{tildeover (F)}(k)| then the variance obeys

${_{f}(F)} = {{{\frac{1}{N^{2}}{\sum\limits_{j}{{f(j)}\left( \frac{F(j)}{{Nf}(j)} \right)^{2}}}} - \left( {(F)} \right)^{2}} \leq {{\frac{4}{N^{2}}\left( {\sum\limits_{k}{\delta_{k}}} \right)\left( {\sum\limits_{j}{{F(j)}}} \right)} + {_{f_{opt}}(F)}}}$

Proof. The proof is a straight forward exercise in the triangleinequality once one uses the fact that |δ_(j)|≤|F(j)|/2 and the factthat 1/(1−|x|)≤1+2|x| for all xϵ[−½, ½]:

$\begin{matrix}\begin{matrix}{{_{f}(F)} = {{\frac{1}{N^{2}}\left( {{\sum\limits_{k}{{F(k)}}} + \delta_{k}} \right)\left( {\sum\limits_{j}\frac{F^{2}(j)}{{{F(j)}} + \delta_{j}}} \right)} - \left( {(F)} \right)^{2}}} \\{\leq {{\frac{1}{N^{2}}\left( {{\sum\limits_{k}{{F(k)}}} + \delta_{k}} \right)\left( {\sum\limits_{j}\frac{F^{2}(j)}{{{F(j)}} - {\delta_{j}}}} \right)} - \left( {(F)} \right)^{2}}} \\{\leq {{\frac{1}{N^{2}}\left( {{\sum\limits_{k}{{F(k)}}} + {\delta_{k}}} \right)\left( {{\sum\limits_{j}{{F(j)}}} + {2{\delta_{j}}}} \right)} - \left( {(F)} \right)^{2}}} \\{= {{\frac{1}{N^{2}}\left( {\sum\limits_{k}{\delta_{k}}} \right)\left( {{\sum\limits_{j}{{F(j)}}} + {2{\delta_{j}}}} \right)} +}} \\{{{\frac{1}{N^{2}}\left( {\sum\limits_{k}{{F(k)}}} \right)\left( {2{\sum\limits_{j}{\delta_{j}}}} \right)} + \left( {\left( {F} \right)} \right)^{2} - \left( {(F)} \right)^{2}}} \\{\leq {{\frac{4}{N^{2}}\left( {\sum\limits_{k}{\delta_{k}}} \right)\left( {\sum\limits_{j}{{F(j)}}} \right)} + {{_{f_{opt}}(F)}.}}}\end{matrix} & (55)\end{matrix}$

This bound is tight in the sense that as max_(k)|δ_(k)|→0 the upperbound on the variance converges to (

(|F|))²−(

(F))² which is the optimal attainable variance.

In applications such as quantum chemistry simulation, what one wants todo is minimize the variance in Lemma 2. This minimum variance could beattained by choosing ƒ(j)∝|

ψ|H_(j)|ψ

|. However, the task of computing such a functional is atleast as hardas solving the eigenvalue estimation problem that one wants to tackle.The natural approach to take is to take inspiration from Lemma 10 andinstead take ƒ(j)∝|

{tilde over (ψ)}|H_(j)|{tilde over (ψ)}

| where |{tilde over (ψ)}

is an efficiently computable ansatz state such as a CISD state. Inpractice, however, the importance of a given term may not be entirelypredicted by the ansatz prediction. In which case a hedging strategy canbe used wherein for some ρϵ[0, 1], ƒ(j)∝(1−ρ)

H_(j)

+ρ∥H_(j)∥. This strategy allows you to smoothly interpolate betweenimportance dictated by the magnitude of the Hamiltonian terms as well asthe expectation value in the surrogate for the groundstate.

VIII. Numerical Results

This disclosure has shown that it is possible to use iterative phaseestimation using a randomized Hamiltonian. To show how effective exampleembodiments can be, two examples of diatomic molecules, dilithium andhydrogen chloride, are considered. In both cases, the molecules areprepared in a minimal STO6G basis and use CISD states found byvariationally minimimizing the groundstate energy over all states within2 excitations away from the Hartree Fock state. One can then randomlysample Hamiltonians terms and then examine several quantities ofinterest including the average groundstate energy, the variance in thegroundstate energies and the average number of terms in the Hamiltonian.Interestingly, one can also look at the number of qubits present in themodel. This can differ because some randomly sampled Hamiltonians willactually choose terms in the Hamiltonian that do not couple with theremainder of the system. In these cases, the number of qubits requiredto represent the state can in fact be lower than the total number thatwould be ordinarily expected.

One can see in FIGS. 2-5 and FIGS. 6-9 that the estimates of the groundstate energy varies radically with the degree of hedging used. It isfound that if ρ=1 for both cases then in all cases, one has a very largevariance in the groundstate energy, as expected since importancesampling has very little impact in that case. Conversely, it is foundthat is one takes ρ=0, one can maximally privilege the importance ofHamiltonian terms from the CISI) state which leads to very concisemodels but with shifts in groundstate energies that are on the order of1 Ha for even 10⁷ randomly selected terms (some of which may beduplicates). If one insteads use a modest amount of hedging (ρ=2×10⁻⁵)then one notices that the shift in the ground state energy is minimizedassuming that a shift in energy of 10%; of chemical accuracy or 0.1 mHais acceptable for Hamiltonian truncation error. For dilithium, thisrepresents a 30% reduction in the number of terms in the Hamiltonian;whereas for HCl this reduces the number of terms in the Hamiltonian by afactor of 3. Since the cost of a Trotter-Suzuki simulation of chemistryscales super-linearly with the number of terms in the Hamiltonian thisconstitutes a substantial reduction in the complexity.

One can also note that for the case of dilithium, the number of qubitsneeded to perform the simulation varied over the different runs. Incontrast Chlorine showed no such behavior. This difference arises fromthe fact that dilithium requires six electrons that reside in 20 spinorbitals. Hydrogen Chloride consists of eighteen electrons and theexample Fock space consists of 20 spin orbitals also. As a result,nearly every spin orbital will be relevant in that which explains whythe number of spin orbitals needed to express dilithium to a fixeddegree of precision changes whereas it does not for HCl. Thisillustrates that embodiments of the disclosed randomization procedurecan be used to help select an active space for a simulation on the flyas the precision needed in the Hamiltonian increases through a phaseestimation procedure.

FIGS. 2-9 comprise graphs 200, 300, 400, 500, 600, 700, 800, and 900that show the average ground energy shift (compared to unsampledHamiltonian), variance in ground energies over sampled Hamiltonians,average qubit requirement, and average number of terms in sampledHamiltonians for Li₂, as a function of number of samples taken togenerate the Hamiltonian and the value of the parameter ρ. A term in theHamiltonian H_(α) is sampled with probability ρ_(α)∝(1−ρ)

H_(α)

+ρ∥H_(α)∥, where the expectation value is taken with the CISD state.

IX. Example Embodiments

In this section, example methods for performing the disclosed technologyare disclosed. The particular embodiments described should not beconstrued as limiting, as the disclosed method acts can be performedalone, in different orders, or at least partially simultaneously withone another. Further, any of the disclosed methods or method acts can beperformed with any other methods or method acts disclosed herein.

FIG. 10 is a flow chart 1000 showing an example method for implementingan importance sampling simulation method according to an embodiment ofthe disclosed technology.

FIG. 11 is a flow chart 1100 showing an example method for performing aquantum simulation using adaptive Hamiltonian randomization.

FIG. 16 is a flow chart 1600 showing an example method for performing aquantum simulation using adaptive Hamiltonian randomization.

At 1610, a Hamiltonian to be computed by the quantum computer device isinputted.

At 1612, a number of Hamiltonian terms in the Hamiltonian is reducedusing randomization within a phase estimation algorithm.

At 1614, a quantum circuit description for the Hamiltonian is outputwith the reduced number of Hamiltonian terms.

In certain embodiments, the reducing comprises selecting one or morerandom Hamiltonian terms based on an importance function; reweightingthe selected random Hamiltonian terms based on an importance of each ofthe selected Hamiltonian random terms; and generating the quantumcircuit description using the reweighted random terms. Some embodimentsfurther comprise implementing, in the quantum computing device, aquantum circuit as described by the quantum circuit description; andmeasuring a quantum state of the quantum circuit. Still furtherembodiments comprise re-performing the method based on results from themeasuring (e.g., using an iterative process). In some embodiments, theiterative process comprises computing a desired precision value for theHamiltonian; computing a standard deviation for the Hamiltonian based onresults from the implementing and measuring; and comparing the desiredprecision value to the standard deviation. Some embodiments furthercomprise changing an order of the Hamiltonian terms based on thereducing. Certain embodiments further comprise applying importancefunctions to terms of the Hamiltonian in a ground state; and selectingone or more random Hamiltonian terms based at least in part on theimportance functions. Some embodiments comprise using importancesampling based on a variational approximation to a groundstate. Certainembodiments further comprise using adaptive Bayesian methods to quantifya precision needed for the Hamiltonian given an estimate of the currentuncertainty in an eigenvalue.

Other embodiments comprise one or more computer-readable media storingcomputer-executable instructions, which when executed by a computercause the computer to perform a method comprising inputting aHamiltonian to be computed by the quantum computer device; reducing anumber of Hamiltonian terms in the Hamiltonian using randomizationwithin a phase estimation algorithm; and outputting a quantum circuitdescription for the Hamiltonian with the reduced number of Hamiltonianterms.

The method can comprise selecting one or more random Hamiltonian termsbased on an importance function: reweighting the selected randomHamiltonian terms based on an importance of each of the selectedHamiltonian random terms; and generating the quantum circuit descriptionusing the reweighted random terms. The method can further comprisecausing a quantum circuit as described by the quantum circuitdescription to implemented by the quantum computing device; andmeasuring a quantum state of the quantum circuit. The method can furthercomprise computing a desired precision value for the Hamiltonian;computing a standard deviation for the Hamiltonian based on results fromthe implementing and measuring: comparing the desired precision value tothe standard deviation; and re-performing the reducing based on a resultof the comparing.

Another embodiment is a system, comprising a quantum computing system;and a classical computing system configured to communicate with andcontrol the quantum computing system. In such embodiments, the classicalcomputing system is further configured to: input a Hamiltonian to becomputed by the quantum computer device; reduce a number of Hamiltonianterms in the Hamiltonian using randomization within an iterative phaseestimation algorithm; and output a quantum circuit description for theHamiltonian with the reduced number of Hamiltonian terms. The classicalcomputing system can be further configured to: select one or more randomHamiltonian terms based on an importance function; reweight the selectedrandom Hamiltonian terms based on an importance of each of the selectedHamiltonian random terms; and generate the quantum circuit descriptionusing the reweighted random terms. The classical computing system can befurther configured to cause a quantum circuit as described by thequantum circuit description to be implemented by the quantum computingdevice; and measure a quantum state of the quantum circuit. Stillfurther, the classical computing system can be further configured tocompute a desired precision value for the Hamiltonian; compute astandard deviation for the Hamiltonian based on results from theimplementing and measuring; compare the desired precision value to thestandard deviation; and re-perform the reducing based on a result of thecomparing. In still further embodiments, the classical computing systemcan be further configured such that, as part of the randomization, oneor more unnecessary qubits are omitted.

X. Example Computing Environments

FIG. 12 illustrates a generalized example of a suitable classicalcomputing environment 1200 in which aspects of the described embodimentscan be implemented. The computing environment 1200 is not intended tosuggest any limitation as to the scope of use or functionality of thedisclosed technology, as the techniques and tools described herein canbe implemented in diverse general-purpose or special-purposeenvironments that have computing hardware.

With reference to FIG. 12, the computing environment 1200 includes atleast one processing device 1210 and memory 1220. In FIG. 12, this mostbasic configuration 1230 is included within a dashed line. Theprocessing device 1210 (e.g., a CPU or microprocessor) executescomputer-executable instructions. In a multi-processing system, multipleprocessing devices execute computer-executable instructions to increaseprocessing power. The memory 1220 may be volatile memory (e.g.,registers, cache, RAM, DRAM, SRAM), non-volatile memory (e.g., ROM,EEPROM, flash memory), or some combination of the two. The memory 1220stores software 1280 implementing tools for performing any of thedisclosed techniques for operating a quantum computer to performHamiltonian randomization as described herein. The memory 1220 can alsostore software 1280 for synthesizing, generating, or compiling quantumcircuits for performing any of the disclosed techniques.

The computing environment can have additional features. For example, thecomputing environment 1200 includes storage 1240, one or more inputdevices 1250, one or more output devices 1260, and one or morecommunication connections 1270. An interconnection mechanism (notshown), such as a bus, controller, or network, interconnects thecomponents of the computing environment 120). Typically, operatingsystem software (not shown) provides an operating environment for othersoftware executing in the computing environment 1200, and coordinatesactivities of the components of the computing environment 1200.

The storage 1240 can be removable or non-removable, and includes one ormore magnetic disks (e.g., hard drives), solid state drives (e.g., flashdrives), magnetic tapes or cassettes, CD-ROMs, DVDs, or any othertangible non-volatile storage medium which can be used to storeinformation and which can be accessed within the computing environment1200. The storage 1240 can also store instructions for the software 1280implementing any of the disclosed techniques. The storage 1240 can alsostore instructions for the software 1280 for generating and/orsynthesizing any of the described techniques, systems, or quantumcircuits.

The input device(s) 1250 can be a touch input device such as a keyboard,touchscreen, mouse, pen, trackball, a voice input device, a scanningdevice, or another device that provides input to the computingenvironment 1200. The output device(s) 1260 can be a display device(e.g., a computer monitor, laptop display, smartphone display, tabletdisplay, netbook display, or touchscreen), printer, speaker, or anotherdevice that provides output from the computing environment 1200.

The communication connection(s) 1270 enable communication over acommunication medium to another computing entity. The communicationmedium conveys information such as computer-executable instructions orother data in a modulated data signal. A modulated data signal is asignal that has one or more of its characteristics set or changed insuch a manner as to encode information in the signal. By way of example,and not limitation, communication media include wired or wirelesstechniques implemented with an electrical, optical, RF, infrared,acoustic, or other carrier.

As noted, the various methods and techniques for performing Hamiltonianrandomization, for controlling a quantum computing device, to performcircuit design or compilation/synthesis as disclosed herein can bedescribed in the general context of computer-readable instructionsstored on one or more computer-readable media. Computer-readable mediaare any available media (e.g., memory or storage device) that can beaccessed within or by a computing environment. Computer-readable mediainclude tangible computer-readable memory or storage devices, such asmemory 1220 and/or storage 1240, and do not include propagating carrierwaves or signals per se (tangible computer-readable memory or storagedevices do not include propagating carrier waves or signals per se).

Various embodiments of the methods disclosed herein can also bedescribed in the general context of computer-executable instructions(such as those included in program modules) being executed in acomputing environment by a processor. Generally, program modules includeroutines, programs, libraries, objects, classes, components, datastructures, and so on, that perform particular tasks or implementparticular abstract data types. The functionality of the program modulesmay be combined or split between program modules as desired in variousembodiments. Computer-executable instructions for program modules may beexecuted within a local or distributed computing environment.

An example of a possible network topology 1300 (e.g., a client-servernetwork) for implementing a system according to the disclosed technologyis depicted in FIG. 13. Networked computing device 1320 can be, forexample, a computer running a browser or other software connected to anetwork 1312. The computing device 1320 can have a computer architectureas shown in FIG. 12 and discussed above. The computing device 1320 isnot limited to a traditional personal computer but can comprise othercomputing hardware configured to connect to and communicate with anetwork 1312 (e.g., smart phones, laptop computers, tablet computers, orother mobile computing devices, servers, network devices, dedicateddevices, and the like). Further, the computing device 1320 can comprisean FPGA or other programmable logic device. In the illustratedembodiment, the computing device 1320 is configured to communicate witha computing device 1330 (e.g., a remote server, such as a server in acloud computing environment) via a network 1312. In the illustratedembodiment, the computing device 1320 is configured to transmit inputdata to the computing device 1330, and the computing device 1330 isconfigured to implement a technique for controlling a quantum computingdevice to perform any of the disclosed embodiments and/or a circuitgeneration/compilation/synthesis technique for generating quantumcircuits for performing any of the techniques disclosed herein. Thecomputing device 1330 can output results to the computing device 1320.Any of the data received from the computing device 1330 can be stored ordisplayed on the computing device 1320 (e.g., displayed as data on agraphical user interface or web page at the computing devices 1320). Inthe illustrated embodiment, the illustrated network 1312 can beimplemented as a Local Area Network (LAN) using wired networking (e.g.,the Ethernet IEEE standard 802.3 or other appropriate standard) orwireless networking (e.g. one of the IEEE standards 802.11a, 802.11b,802.11g, or 802.11n or other appropriate standard). Alternatively, atleast part of the network 1312 can be the Internet or a similar publicnetwork and operate using an appropriate protocol (e.g., the HTTPprotocol).

Another example of a possible network topology 1400 (e.g., a distributedcomputing environment) for implementing a system according to thedisclosed technology is depicted in FIG. 14. Networked computing device1420 can be, for example, a computer running a browser or other softwareconnected to a network 1112. The computing device 1420 can have acomputer architecture as shown in FIG. 12 and discussed above. In theillustrated embodiment, the computing device 1420 is configured tocommunicate with multiple computing devices 1430, 1431, 1432 (e.g.,remote servers or other distributed computing devices, such as one ormore servers in a cloud computing environment) via the network 1412. Inthe illustrated embodiment, each of the computing devices 1430, 1431,1432 in the computing environment 1400 is used to perform at least aportion of the Hamiltonian randomization technique and/or at least aportion of the technique for controlling a quantum computing device toperform any of the disclosed embodiments and/or a circuitgeneration/compilation/synthesis technique for generating quantumcircuits for performing any of the techniques disclosed herein. In otherwords, the computing devices 1430, 1431, 1432 form a distributedcomputing environment in which aspects of the techniques for performingany of the techniques as disclosed herein and/or quantum circuitgeneration/compilation/synthesis processes are shared across multiplecomputing devices. The computing device 1420 is configured to transmitinput data to the computing devices 1430, 1431, 1432, which areconfigured to distributively implement such as process, includingperformance of any of the disclosed methods or creation of any of thedisclosed circuits, and to provide results to the computing device 1420.Any of the data received from the computing devices 1430, 1431, 1432 canbe stored or displayed on the computing device 1420 (e.g., displayed asdata on a graphical user interface or web page at the computing devices1420). The illustrated network 1412 can be any of the networks discussedabove with respect to FIG. 13.

With reference to FIG. 15, an exemplary system for implementing thedisclosed technology includes computing environment 1500. In computingenvironment 1500, a compiled quantum computer circuit description(including quantum circuits for performing any of the disclosedtechniques as disclosed herein) can be used to program (or configure)one or more quantum processing units such that the quantum processingunit(s) implement the circuit described by the quantum computer circuitdescription.

The environment 1500 includes one or more quantum processing units 1502and one or more readout device(s) 1508. The quantum processing unit(s)execute quantum circuits that are precompiled and described by thequantum computer circuit description. The quantum processing unit(s) canbe one or more of, but are not limited to: (a) a superconducting quantumcomputer; (b) an ion trap quantum computer; (c) a fault-tolerantarchitecture for quantum computing; and/or (d) a topological quantumarchitecture (e.g., a topological quantum computing device usingMajorana zero modes). The precompiled quantum circuits, including any ofthe disclosed circuits, can be sent into (or otherwise applied to) thequantum processing unit(s) via control lines 1506 at the control ofquantum processor controller 1520. The quantum processor controller (QPcontroller) 1520 can operate in conjunction with a classical processor1510 (e.g., having an architecture as described above with respect toFIG. 12) to implement the desired quantum computing process. In theillustrated example, the QP controller 1520 further implements thedesired quantum computing process via one or more QP subcontrollers 1504that are specially adapted to control a corresponding one of the quantumprocessor(s) 1502. For instance, in one example, the quantum controller1520 facilitates implementation of the compiled quantum circuit bysending instructions to one or more memories (e.g., lower-temperaturememories), which then pass the instructions to low-temperature controlunit(s) (e.g., QP subcontroller(s) 1504) that transmit, for instance,pulse sequences representing the gates to the quantum processing unit(s)1502 for implementation. In other examples, the QP controller(s) 1520and QP subcontroller(s) 1504 operate to provide appropriate magneticfields, encoded operations, or other such control signals to the quantumprocessor(s) to implement the operations of the compiled quantumcomputer circuit description. The quantum controller(s) can furtherinteract with readout devices 1508 to help control and implement thedesired quantum computing process (e.g., by reading or measuring outdata results from the quantum processing units once available, etc.)

With reference to FIG. 15, compilation is the process of translating ahigh-level description of a quantum algorithm into a quantum computercircuit description comprising a sequence of quantum operations orgates, which can include the circuits as disclosed herein (e.g., thecircuits configured to perform one or more of the procedures asdisclosed herein). The compilation can be performed by a compiler 1522using a classical processor 1510 (e.g., as shown in FIG. 12) of theenvironment 1500 which loads the high-level description from memory orstorage devices 1512 and stores the resulting quantum computer circuitdescription in the memory or storage devices 1512.

In other embodiments, compilation and/or verification can be performedremotely by a remote computer 1560 (e.g., a computer having a computingenvironment as described above with respect to FIG. 12) which stores theresulting quantum computer circuit description in one or more memory orstorage devices 1562 and transmits the quantum computer circuitdescription to the computing environment 1500 for implementation in thequantum processing unit(s) 1502. Still further, the remote computer 1500can store the high-level description in the memory or storage devices1562 and transmit the high-level description to the computingenvironment 1500 for compilation and use with the quantum processor(s).In any of these scenarios, results from the computation performed by thequantum processor(s) can be communicated to the remote computer afterand/or during the computation process. Still further, the remotecomputer can communicate with the QP controller(s) 1520 such that thequantum computing process (including any compilation, verification, andQP control procedures) can be remotely controlled by the remote computer1560. In general, the remote computer 1560 communicates with the QPcontroller(s) 1520, compiler/synthesizer 1522, and/or verification tool1523 via communication connections 1550.

In particular embodiments, the environment 1500 can be a cloud computingenvironment, which provides the quantum processing resources of theenvironment 1500 to one or more remote computers (such as remotecomputer 1560) over a suitable network (which can include the internet).

XI. Concluding Remarks

This application has shown that iterative phase estimation is moreflexible than it previously may have been thought and that the number ofterms in the Hamiltonian be randomized at each step of iterative phaseestimation without substantially contributing to the underlying varianceof an unbiased estimator of the eigenphase. It was further shownnumerically that by using such strategies for sub-sampling theHamiltonian terms that one can perform a simulation using fewerHamiltonian terms than would be necessary for traditional approachesrequire. These reductions in the number of terms directly impact thecomplexity of Trotter-Suzuki based simulation and indirectly impactqubitization and truncated Taylor series simulation methods because theyalso reduce the 1-norm of the vector of Hamiltonian terms.

Having described and illustrated the principles of the disclosedtechnology with reference to the illustrated embodiments, it will berecognized that the illustrated embodiments can be modified inarrangement and detail without departing from such principles. Forinstance, elements of the illustrated embodiments shown in software maybe implemented in hardware and vice-versa. Also, the technologies fromany example can be combined with the technologies described in any oneor more of the other examples. It will be appreciated that proceduresand functions such as those described with reference to the illustratedexamples can be implemented in a single hardware or software module, orseparate modules can be provided. The particular arrangements above areprovided for convenient illustration, and other arrangements can beused.

1. A method of operating a quantum computing device, comprising:inputting a Hamiltonian to be computed by the quantum computer device;reducing a number of Hamiltonian terms in the Hamiltonian usingrandomization within a phase estimation algorithm; and outputting aquantum circuit description for the Hamiltonian with the reduced numberof Hamiltonian terms.
 2. The method of claim 1, wherein the method isperformed by a classical computer.
 3. The method of claim 1, wherein thereducing comprises: selecting one or more random Hamiltonian terms basedon an importance function; reweighting the selected random Hamiltonianterms based on an importance of each of the selected Hamiltonian randomterms; generate the quantum circuit description using the reweightedrandom terms.
 4. The method of claim 3, further comprising:implementing, in the quantum computing device, a quantum circuit asdescribed by the quantum circuit description; measuring a quantum stateof the quantum circuit.
 5. The method of claim 4, further comprisingre-performing the method of claim 4 based on results from the measuring.6. The method of claim 5, wherein the re-performing is performed basedon an interative process.
 7. The method of claim 6, wherein theinterative process comprises: computing a desired precision value forthe Hamiltonian; computing a standard deviation for the Hamiltonianbased on results from the implementing and measuring; and comparing thedesired precision value to the standard deviation.
 8. The method ofclaim 1, further comprising changing an order of the Hamiltonian termsbased on the reducing.
 9. The method of claim 1, further comprising:applying importance functions to terms of the Hamiltonian in a groundstate; and selecting one or more random Hamiltonian terms based at leastin part on the importance functions.
 10. The method of claim 1, furthercomprising: using importance sampling based on a variationalapproximation to a groundstate.
 11. The method of claim 1, furthercomprising: using adaptive Bayesian methods to quantify a precisionneeded for the Hamiltonian given an estimate of the current uncertaintyin an eigenvalue.
 12. One or more computer-readable media storingcomputer-executable instructions, which when executed by a computercause the computer to perform a method, the method comprising: inputtinga Hamiltonian to be computed by the quantum computer device; reducing anumber of Hamiltonian terms in the Hamiltonian using randomizationwithin a phase estimation algorithm; and outputting a quantum circuitdescription for the Hamiltonian with the reduced number of Hamiltonianterms.
 13. The one or more computer-readable media of claim 12, whereinthe method further comprises: selecting one or more random Hamiltonianterms based on an importance function; reweighting the selected randomHamiltonian terms based on an importance of each of the selectedHamiltonian random terms; and generating the quantum circuit descriptionusing the reweighted random terms.
 14. The one or more computer-readablemedia of claim 13, wherein the method further comprises: causing aquantum circuit as described by the quantum circuit description toimplemented by the quantum computing device; and measuring a quantumstate of the quantum circuit.
 15. The one or more computer-readablemedia of claim 14, wherein the method further comprises: computing adesired precision value for the Hamiltonian; computing a standarddeviation for the Hamiltonian based on results from the implementing andmeasuring; comparing the desired precision value to the standarddeviation; and re-performing the reducing based on a result of thecomparing.
 16. A system, comprising: a quantum computing system; and aclassical computing system configured to communicate with and controlthe quantum computing system, the classical computing system beingfurther configured to: input a Hamiltonian to be computed by the quantumcomputer device; reduce a number of Hamiltonian terms in the Hamiltonianusing randomization within an iterative phase estimation algorithm; andoutput a quantum circuit description for the Hamiltonian with thereduced number of Hamiltonian terms.
 17. The system of claim 16, whereinthe classical computing system is further configured to: select one ormore random Hamiltonian terms based on an importance function; reweightthe selected random Hamiltonian terms based on an importance of each ofthe selected Hamiltonian random terms; and generate the quantum circuitdescription using the reweighted random terms.
 18. The system of claim17, wherein the classical computing system is further configured to:cause a quantum circuit as described by the quantum circuit descriptionto be implemented by the quantum computing device; and measure a quantumstate of the quantum circuit.
 19. The system of claim 18, wherein theclassical computing system is further configured to: compute a desiredprecision value for the Hamiltonian; compute a standard deviation forthe Hamiltonian based on results from the implementing and measuring;compare the desired precision value to the standard deviation; andre-perform the reducing based on a result of the comparing.
 20. Thesystem of claim 16, wherein the classical computing system is furtherconfigured to, as part of the randomization, one or more unnecessaryqubits are omitted.